State-Dependent Coefficients in Electrical-Engineering Pedagogy: A Comparative Metrological and Coupling-Theory Audit with a Reserved Paraformer Test Section

Introductory and intermediate electrical-engineering education commonly models fundamental circuit and device parameters, including inductance, capacitance, resistance, permeability, permittivity, conductivity, characteristic impedance, transformer turns ratio, machine constants, amplifier gain, resonant frequency, propagation velocity, and mutual inductance, as numerical constants. While this approximation is valid within the intended small-signal operating regime, it becomes methodologically incomplete when these coefficients exhibit measurable state dependence. This paper presents a comparative audit of thirteen such coefficients by systematically contrasting their textbook formulations with the established engineering literature and interpreting the results through two complementary frameworks: the JCGM GUM-6:2020 measurement-model methodology for omitted effects, and the Heckmann–Nye/Gasik multidomain coupling architecture for multi-axis physical interactions. The analysis demonstrates that mainstream engineering practice routinely exploits state-dependent coefficients without invoking new physical laws, and that relaxing the constant-coefficient assumption naturally introduces physically meaningful terms, including the inductive contribution IdL/dt, the capacitive counterpart VdC/dt, and the mutual-inductance term i2dM/dt. The principal scientific contribution is the development and experimental validation of a unified theoretical and engineering framework for high-power resonant transformers and orthogonal Metglas AMCC-1000 paraformers. The proposed approach combines a new physics-based modal theory of octave (2:1) parametric excitation with simultaneous optimization of magnetic-core resonance, electrical resonance, nonlinear inductance modulation, resonant conductor lengths selected as integer multiples of the operating resonant wavelength, multi-stranded high-frequency Litz-wire windings, resonant capacitor synthesis, and the nonlinear magnetic characteristics of the AMCC-1000 amorphous core. The modal analysis demonstrates how coupled resonant eigenmodes and engineered state-dependent inductance can be used to satisfy the conditions for stable octave parametric excitation. Experimental results obtained from both the symmetric two-leg resonant transformer and the orthogonal paraformer are in close agreement with analytical predictions and numerical simulations, thereby validating both the proposed electromagnetic design methodology and the underlying modal theory.